Commutators in residually finite groups

Shumyatsky, Pavel

doi:10.1007/s11856-011-0027-3

Cited by 10 publications

(2 citation statements)

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“…The general case of profinite groups is dealt with by bounding the nonsoluble length of the group, which enables induction on this length. (We introduced the nonsoluble length in [10], although bounds for nonsoluble length had been implicitly used in various earlier papers, for example, in the celebrated Hall-Higman paper [5], or in Wilson's paper [25]; more recently, bounds for the nonsoluble length were used in the study of verbal subgroups in finite and profinite groups [3,21,22,11].) Finally, the result for compact groups is derived with the use of the structure theorems for compact groups.…”

Section: Definitionmentioning

confidence: 99%

Compact groups in which all elements have countable right Engel sinks

Khukhro

Shumyatsky²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$ .) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

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Section: Definitionmentioning

confidence: 99%

Compact groups in which all elements have countable right Engel sinks

Khukhro

Shumyatsky²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…The general case of profinite groups is dealt with by bounding the nonsoluble length of the group, which enables induction on this length. (We introduced the nonsoluble length in [12], although bounds for nonsoluble length had been implicitly used in various earlier papers, for example, in the celebrated Hall-Higman paper [7], or in Wilson's paper [23]; more recently, bounds for the nonsoluble length were studied in connection with verbal subgroups in finite and profinite groups in [4,6,13,20,22]. )…”

Section: Introductionmentioning

confidence: 99%

Strong conciseness of Engel words in profinite groups

Khukhro

Shumyatsky

2023

Mathematische Nachrichten

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A group word 𝑤 is said to be strongly concise in a class 𝒞 of profinite groups if, for any group 𝐺 in 𝒞, either 𝑤 takes at least continuum many values in 𝐺 or the verbal subgroup 𝑤(𝐺) is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups, as well as that 2-Engel words are strongly concise (but their approach does not seem to generalize to 𝑛-Engel words for 𝑛 > 2). In this paper, we prove that for any 𝑛, the 𝑛-Engel word [… [ [𝑥, 𝑦], 𝑦], … 𝑦] (where 𝑦 is repeated 𝑛 times) is strongly concise in the class of finitely generated profinite groups.

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